A naive attempt is to try and use the vertex cover lower bound graph \(G_{x,y}\)  from [1]} and replace each edge with a vertex of weight 0,
in order to get a new lower bound graph \(H_{x,y}\)  with the same size of solution for \(H^2\)  as in \(G\) .
The issue is that in the graph \(H_{x,y}\) , the number of vertices is \(O(m)\)  where \(m = O(n^{2})\)  is the number of edges in \(G_{x,y}\) . This is a major issue if we want super linear lower bounds as a quadratic lower bound in \(G_{x,y}\)  only gives a linear lower bound for \(H^{2}_{x,y}\) . So instead, we modify the lower bound graph construction of [1]} in a subtle manner to show an \(\Omega (n^2)\)  lower bound for computing exact MWVC in \(G^2\) .
